ON CERTAIN DEVELOPABLE SURFACES. 
268 
[344 
where t is an arbitrary parameter, and the coefficients (a, b,...) are linear functions 
of the coordinates; the equation of the developable is 
Disct. (a, b, l) n = 0, 
the discriminant being taken in regard to the parameter t. Such developable is in 
general of the order 2n — 2, but if the second coefficient b is = 0, or, more generally, 
if it is a mere numerical multiple of a, then a will divide out from the equation, and 
we have a developable of the order 2n — 3: the like property of course exists in regard 
to the last but one, and the last, of the coefficients of the function. We thus obtain 
developables of the orders 4, 5, and 6, sufficiently simple to allow of the actual 
calculation of their Prohessians, and the chief object of the present Memoir is to 
exhibit these Prohessians; but the Memoir contains some other researches in relation 
to the developables in question. 
Quartic Developable, Nos. 1 to 6. 
1. I consider first the developable of the fourth order 
U = a 2 d 2 — 6abcd + 4ac 3 + 4 b 3 d — 3 6 2 c 2 , 
derived from the cubic function (a, b, c, d~$t, l) 3 , and which is in fact the general 
quartic developable. 
2. Taking (a, b, c, d) as coordinates and omitting common numerical factors, the 
first derived functions are 
ad 2 — 3bed + 2c 3 , 
— 3 acd + 6b 2 d — 36c 2 , 
— 3abd -f 6ac 2 — 36 2 c, 
a 2 d — Sabc + 2b 3 , 
(quantities which, if (X, Y, Z, W$t, l) 3 denote the cubicovariant of {a, b, c, d^t, l) 3 , 
are equal to (— W, 3Z, — 3 Y, X) respectively). And the second derived functions are 
cP 
— 3 cd , 
- 
3bd + 6c 2 , 
2 ad — 36c, 
— 3 cd , 
12bd — 3c 2 , 
- 
3 ad — 6 6c, 
— 3ac + 66 2 , 
— 3bd + 6c 2 , 
— 3acZ — 66c, 
12ac — 36 2 , 
— Sab , 
2 ad — 36c, 
— 3ac + 66 2 , 
- 
Sab , 
a 2 
Representing these by 
A, H, 
G, 
L, 
H, B, 
F, 
M, 
G, F, 
G, 
N, 
L, M, 
N, 
P,